1. Frequency Dependence of Quantum Localization in a Periodically Driven System Manabu Machida, Keiji Saito, and Seiji Miyashita Department of Applied Physics, The University of Tokyo

2. Matrices of Gaussian Orthogonal Ensemble (GOE) are real symmetric, and each element of them is a Gaussian distributed random number. GOE Random Matrix E.P. Wigner introduced random matrices to Physics. Wigner, F.J. Dyson, and many other physicists developed random matrix theory.

3. andare independently created GOE random matrices. is fixed at 0.5. Hamiltonian varies. Typical Hamiltonian for complexly interacting systems under an external field.

4. Floquet Theory

5. Energy after nth period: We define, Energy fluctuates around

6. Comparing Saturated ! Solid line Esat is normalized so that the ground state energy is 0 and the energy at the center of the spectrum is 1. with 0.02 0.1 0.2 0.4 1.0

7. as a function of

8. How to understand the localization? (i) Independent Landau-Zener Transitions Wilkinson considered the energy change of a random matrix system when the parameter is swept. M. Wilkinson, J.Phys.A 21 (1988) 4021 M. Wilkinson, Phys.Rev.A 41 (1990) 4645 We assume transitions of states occur at avoided crossings by the Landau-Zener formula, and each transition takes place independently.

9. How to understand the localization? Transition probability Probability of finding the state on the lth level Diffusion equation:

10. The integral on the exponential diverges. Therefore, Quantum interference effect is essential! for any  How to understand the localization? The global transition cannot be understood only by the Landau-Zener transition.

11. The random matrix systemThe Anderson localization In each time interval T, the system evolves by the Floquet operator F. The Hamiltonian which brings about the Anderson localization evolves in the interval T, How to understand the localization? (ii) Analogy to the Anderson Localization

12. : random potential distributed uniformly in the width W :Hamiltonian for the Anderson localization How to understand the localization?

13. Let us introduce in order to study  -dependence of the quantum localization. F. Haake, M. Kus, and R. Scharf, Z.Phys.B 65 (1987) 381 K. Zyczkowski, J.Phys.A 23 (1990) 4427 We count the number of relevant Floquet states in the initial state.

14. One important aspect of the quantum localization

15.  -dependence of Nmin Phenomenologically,

16. Parameters in the phenomenological function of

17. (numerical)

18. : unknown amplitude This fact suggests the local transition probability originates in the Landau-Zener transition.

19. The quantum localization occurs in this random matrix due to the quantum interference effect. On the other hand, the Landau-Zener mechanism still works in the local transitions. To be appeared in J.Phys.Soc.Jpn. 71(2002) Conclusion